We make some remarks about these assumptions.
Assumption REF  is without loss of generality since inequalities can be reformulated as equality constraints by adding squares of additional slack variables, and it is also used in the convergence analyses of existing augmented Lagrangian methods (e.g., see [1]}, [2]}, [3]}).
Assumptions REF  and REF  state that the linear independence constraint qualification (LICQ) and second-order sufficient conditions (SOSC) are satisfied, respectively, at the local solution \((x^*, z^*)\)  of the quadratic penalty formulation (REF ) for all (sufficiently large) \(\theta > 0\) .
Assumption REF  states that LICQ is also satisfied at local solutions of problem (REF ) that correspond to the \(x\) -updates.
As before, these are fairly standard to establish local convergence of augmented Lagrangian methods.
Finally, Assumption REF  is satisfied by any well-behaved NLP solver if \(A_t\)  has full column rank since all variables have an associated proximal term in problem (REF ) in this case; when \(A_t\)  does not have full column rank, the assumption is necessary for convergence and it is implicit in classical analyses of the standard augmented Lagrangian method [1]}.
